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HMMT 十一月 2010 · GEN1 赛 · 第 7 题

HMMT November 2010 — GEN1 Round — Problem 7

专题
Contest Math / 竞赛数学
难度
L3
来源
HMMT

题目详情

英文原题

  1. [ 6 ] What is the remainder when (1 + x ) is divided by 1 + x + x ?
解析

英文解析

  1. [ 6 ] What is the remainder when (1 + x ) is divided by 1 + x + x ?
    Answer: 1 We use polynomial congruence mod 1 + x + x to find the desired remainder. Since 2
    2 3 3 2
    x + x + 1 | x − 1, we have that x ≡ 1 (mod 1 + x + x ). Now:
    2010 2 2010 2
    (1 + x ) ≡ ( − x ) (mod 1 + x + x )
    4020 2
    ≡ x (mod 1 + x + x )
    3 1340 2
    ≡ ( x ) (mod 1 + x + x )
    1340 2
    ≡ 1 (mod 1 + x + x )
    ≡ 1 (mod 1 + x + x )2
    Thus, the answer is 1.